Don’t ask me to be more precise, but I’m actually rather glad that Hatcher has written his noted books; there’s a frame of mind, or a kind of puzzle, in which I’ve found his AT to be very helpful. His proof of skeletal compression comes to mind — that given skeleta for $X,Y$ and a map $f : X \to Y$, $f$ is homotopic to a union of filtered maps $ f_n : X_n \to Y_n$. I don’t, however, (and you can probably guess why) enjoy working through explicitly cellular proofs — the sort that goes “here’s a sneaky model for the particular space we want to study, and here is a mess of algebra showing that this next sneaky cellwise map does the right thing”. His run-through of constructing the Steenrod operations and proving the Adem Relations is a fine example of such a slog.

The something I have in mind, I think I have a cellular argument for; but I’m not sure if it’s *quite* right, and it’s nothing as careful as Hatcher’s arguments seem to be. How I wish I could phrase it in some other way, alas!

By induction on $k$ (if you like, or keep reading) there’s a map (actually, about two pointed maps) $ (P_{(2)}^2)^{\wedge k} \to P_{(2)}^{2k}$ inducing injections on $H^*(-,R)$ for all $R$; thinking cellwise on the other hand, the cohomology of $(P_{(2)}^2)^{\wedge k}$ implies (via something due to Whitehead, the smash being simply-connected) that we can choose a cellular presentation of $(P_{(2)}^2)^{\wedge k}$ having exactly one “generator” cell $e_{2k-1}$ in dimension ${2k}-1$, and a single cell (a “relator”) $e_{2k}$ in dimension $2k$ attached by a map $\varphi$ inducing $H^{2k-1} ( X_{2k-1} ) \to H^{2k-1} (\partial e_{2k}) $, a map of index $2$, with image generated your favourite class supported on $ [ e_{2k-1} ] $, where $X_{2k-1}$ is the relevant skeleton.

The map $ (P_{(2)}^2)^{\wedge k}\to P_{(2)}^{2k} $ can then be described as: map $X_{2k-2}$ to the basepoint, map the “relator” cells in dimension $2k-1$ to the basepoint; and what’s left over is a map of $P_{(2)}^{2k}$ to itself. And the reason we have phrased the argument in this way is: there is a map $ P_{(2)}^{2k} \to (P_{(2)}^2)^{\wedge k}$ that restricts to $\sph^{2k-1}$ as $ \varphi \circ \underline{2}$, because $\varphi$ is nulhomotopic in $(P_{(2)}^2)^{\wedge k}$; meanwhile, chasing through the construction, one has that the composite $P_{(2)}^{2k} \to P_{(2)}^{2k}$ is the degree-two map. I think. Trying to write the whole thing more carfully is such that, well, I don’t trust me, yet.

Just to close this note, there’s an extremal character to getting the degree-two map as a composite of this sort; one can’t have $P_{(2)}^{2k}$ a *retract* of $(P_{(2)}^2)^{\wedge k}$, because otherwise you could concoct a space $A$ and a map $ A \vee P^{2k} \to (P_{(2)}^2)^{\wedge k} $ inducing isomorphisms in homology, with $A$ being $2k-1$-dimensional, whence (everything being simply-connected) our map becomes an equivalance; but $(P_{(2)}^2)^{\wedge k}$ has a nontrivial ${Sq}^k$, which is inconsistent with such a splitting. Another time, if it turns out to be useful and true, I might tell some more stories about why we want this factorization of $\underline{2} : P^{2k}$.